Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

Mechanical Engineering: Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

The document Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering is a part of the Mechanical Engineering Course Heat Transfer.
All you need of Mechanical Engineering at this link: Mechanical Engineering

Heat Transfer from Extended Surface (Fin)

  • A fin is a surface that extends from an object to increase the rate of heat transfer to or from the environment by increase convection.
  • Adding a fin to an object increases the surface area and can sometimes be an economical solution to heat transfer problems.
  • Finned surfaces are commonly used in practice to enhance heat transfer. In the analysis of the fins, we consider steady operation with no heat generation in the fin.
  • We also assume that the convection heat transfer coefficient h to be constant and uniform over the entire surface of the fin.
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
  • The rate of heat transfer from a solid surface to atmosphere is given by Q = hA ∆ T where, h and ∆T are not controllable.
  • So, to increase the value of Q surface area should be increased. The extended surface which increases the rate of heat transfer is known as fin.

Generalized Equation for Fin Rectangular Fin

Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
Where Ac and As are cross-sectional and surface area:
And θ(x) = t(x)–ta 

Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

Heat balance equation if Ac constant and As ∞ P(x) linear:
Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

  • General equation of 2nd order: θ = c1emx + c2e–mx
  • Heat dissipation can take place on the basis of three cases.

Case 1: Heat Dissipation from an Infinitely Long Fin (l → ∞):

  • In such a case, the temperature at the end of Fin approaches to surrounding fluid temperature ta as shown in figure. The boundary conditions are given below
  • At x = 0, t = t0 : θ = t0 – ta = θ0
  • At x = L→ ∞: t = ta, θ = 0
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
    θ = θ0 e–mx 

Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

  • Heat transfer by conduction at base:
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

Case 2: Heat Dissipation from a Fin Insulated at the End Tip:

  • Practically, the heat loss from the long and thin film tip is negligible, thus the end of the tip can be, considered as insulated.
  • At x = 0, t = t0 and θ = t0 – ta = θ0 
    at x = l , Q = 0 i.e, dθ/dx = 0
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

Case 3: Heat Dissipation from a Fin loosing Heat at the End Tip:

  • The boundary conditions are given below.
  • At x = 0, t = t0 and θ = θ0
  • At x = l
    Qconduction = Qconvection 
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

Fin Efficiency

Fin efficiency is given by:
Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

  • If l → ∞ (infinite length of fin):
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
  • If fin is with insulated tip:
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
  • If finite length of fin:
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

Note: The following must be noted for a proper fin selection:  

  • The longer the fin, the larger the heat transfer area and thus the higher the rate of heat transfer from the fin.
  • The larger the fin, the bigger the mass, the higher the price, and larger the fluid friction.
  • The fin efficiency decreases with increasing fin length because of the decrease in fin temperature with length.

Fin Effectiveness 

  • The performance of fins is judged on the basis of the enhancement in heat transfer relative to the no‐fin case, and expressed in terms of the fin effectiveness:
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
  • For a sufficiently long fin of uniform cross‐section Ac, the temperature at the tip of the fin will approach the environment temperature, T. By writing energy balance and solving the differential equation, one finds:
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering
    where Ac is the cross‐sectional area, x is the distance from the base, and p is perimeter.  The effectiveness becomes:
    Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering

To increase fins effectiveness, one can conclude:  

  • The thermal conductivity of the fin material must be as high as possible     
  • The ratio of perimeter to the cross‐sectional area p/Ac should be as high as possible 
  • The use of fin is most effective in applications that involve low convection heat transfer coefficient, i.e. natural convection.
The document Heat Transfer Through Fins Notes | Study Heat Transfer - Mechanical Engineering is a part of the Mechanical Engineering Course Heat Transfer.
All you need of Mechanical Engineering at this link: Mechanical Engineering

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